The optical response of single-walled carbon nanotubes is dominated by exciton states with unusually large binding energies. We show that screening in semiconducting tubes enhances rather than reduces the electron-hole interaction for separations larger than the tube diameter. This "antiscreening" region deepens the relative energy level of the higher exciton states yielding unconventional excitation spectra. The effect explains the discrepancy in the current experimentally extrapolated exciton binding energies (deduced using conventional model spectra) and those obtained from ab initio calculations on isolated tubes.